From Axioms to Life: The PDL-V Conjecture and the Hierarchical Replication of C1–C4 (Projective Dynamic Logo Framework, Document DL01)

The Projective Dynamic Logo (PDL) framework derives the fundamental constants of physics — the proton mass, Newton's constant, the cosmological constant — from four combinatorial axioms C1–C4 on finite signed graphs, without presupposing spacetime, particles, or fields. Documents D01–D53 establish this derivation as a closed causal chain with no free parameter beyond the quark mass difference Δm_iso. The present document opens the DL series, a new programme that asks whether the same four axioms, applied hierarchically to composite closures via a lifting operator L, force the emergence of life and consciousness as structural necessities rather than as contingent accidents. Three formal objects are introduced: the lifting operator L, which constructs composite closures from lower-level ones via the (A)∧(B) coupling criterion already operative in the foundational corpus; the constraint-structure operator S(Γ), which maps a closure to the signed graph of its own coherence constraints; and the self-representation relation R^k(Γ), which measures the degree to which a closure encodes its own constraint structure. The key computational result is that S(K₄) ≅ K₄ unconditionally for all 8 coherent configurations of K₄ — the minimal admissible closure is the unique fixed point of S among complete graphs. Weak self-representation R¹_weak is forced at 100% for all coherent closures with n ≥ 4 vertices (verified for n = 4, 5, 6, 7 by exhaustive enumeration). The optimal inter-generational variation δ* = 1/2 is identified as the half-pulsation of K₄. The exact rarity law f(n) = 2^(n+1−C(n,2)) establishes the existence of a finite thermodynamic ceiling n*_max as a combinatorial necessity. The PDL-V conjecture is stated in four parts: the existence of a life threshold n_life forced by C4, the selection of actively self-representing closures by C4 at the lineage scale, the logical necessity of the birth–death cycle as a C4 optimum, and the finite thermodynamic ceiling n_max. Falsification criteria are provided explicitly for each part. The synaptic mechanism of the neural network is presented as a mechanistic illustration of the framework: the synapse as a signed edge, Hebbian plasticity as C4 at the neural level, and the default mode network as a candidate R²-subgraph. These are structural parallels, not formal derivations; the latter are the programme of subsequent DL documents. All claims are stratified by epistemic status. The PDL-V conjecture is not yet a theorem.